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On the Topological Aspects of the Theory of Represented Spaces∗

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On the Topological Aspects of the Theory of Represented Spaces∗ On the topological aspects of the theory of represented spaces∗ Arno Pauly Clare College University of Cambridge, United Kingdom [email protected] Represented spaces form the general setting for the study of computability derived from Turing ma- chines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by ESCARDO´ ,SCHRODER¨ , and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and – closely linked to these – properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property. 1 Introduction ∗ Just as numberings provide a tool to transfer computability from N (or f0;1g ) to all sorts of countable structures; representations provide a means to introduce computability on structures of the cardinality of the continuum based on computability on Cantor space f0;1gN. This is of course essential for com- n m k n m putable analysis [66], dealing with spaces such as R, C (R ;R ), C (R ;R ) or general Hilbert spaces [8]. Computable measure theory (e.g. [65, 58, 44, 30, 21]), or computability on the set of countable ordinals [39, 48] likewise rely on representations as foundation. Essentially, by equipping a set with a representation, we arrive at a represented space – to some extent1 the most general structure carrying a notion of computability that is derived from Turing ma- chines2. Any represented space provides in a canonic way also function spaces and spaces of certain subsets. While the usefulness of these derived spaces will vary depending on the area of application, their development can be done while abstracting away from the specifics of the original represented space – in fact, such details have traditionally obfuscated the core proof ideas. Closely linked to the var- ious subset spaces, also a few properties of spaces defined in terms of computability of certain maps are studied. Among these, special attention shall be drawn to admissibility, which has often been advocated as a criterion for well-behaved representations in computable analysis [66, 56]. arXiv:1204.3763v3 [math.LO] 2 Mar 2015 ∗Prior versions of this manuscript were titled Compactness and Separation for Represented Spaces and A new introduction to the theory of represented spaces. 1The notion of a multi-representation [55, 68] goes beyond representations. A multi-representation of a set X is a right-total relation d ⊆ f0;1gN × X, relating codes to the encoded elements. The distinction to representation is that a code can stand for more than one element here. Several common examples of multi-representations have the additional feature that 9p 2 f0;1gNd(p;x) ^ d(p;y) defines an equivalence relation on X – hence, we can conceive of the multi-representation as an ordinary representation of the induced equivalence classes. Often we even have a canonization operation available; such as the saturation used in Section 5 and the closure used in Section 7 to obtain represented spaces. A very different example (communicated to the author by BAUER) is the multirepresentation 2 : f0;1gN ⇒ P(f0;1gN) n f/0g . Examples of this kind, however, are beyond the scope of the paper. 2Algebraic computation models [6] as abstract generalizations of register machines are a completely different thing. 2 Theory of Represented Spaces 2 Connections to the literature It is well-known that many representations in computable analysis can be characterized by extremal properties – the prototypic example being that the standard representation of the space of continuous functions carries exactly as much information as needed to make function evaluation computable. In fact, this condition not only characterizes the representation, but also the set itself: Whenever a function is contained in a function space admitting computable function evaluation, it is computable w.r.t. some oracle – its name – hence, it is continuous. Consequently, rather than speaking merely about representa- tions (of fixed sets) being characterized such, we should consider represented spaces – the combination of a set and a representation of it – as the fundamental objects of the characterization results3. In many characterization results in the literature the representation is defined explicitly first, then the extremal property is proven. However, as a consequence of the UTM theorem, this is unnecessary: The condition itself defines the represented space up to computable isomorphisms. In fact, specifying details of standard representations beyond their characterizing property often complicates proofs of basic results, and can obfuscate the algorithmic ideas. Developing the basic theory of represented spaces based on extremal characterizations amounts to an instantiation of ESCARDO´ ’s synthetic topology [25, 28] with the category of represented spaces. As this category is particularly well-behaved, we obtain a very concise picture, and can prove a few more results on compactness and separation than available in the generic setting. Similarly, there are many parallels to TAYLOR’s abstract stone duality [60, 61], in comparison we again sacrifice generality for simplicity of presentation and strength of some results. Most definitions and many results about them were already formulated by SCHRODER¨ [54] before the advent of synthetic topology though. In this development, spaces were always required to satisfy some form of admissibility4. For our purposes, these restrictions are mostly immaterial, and will be dropped. Note that admissibility (w.r.t. a topology) turns out to be a feature of synthetic topology in general, demarking those spaces actually fully understandable in terms of their (internal) topology. For represented spaces, this can be seen as (slightly) generalizing [56]. The PhD thesis of LIETZ [40] also contains relevant results on admissibility, in particular [40, Theorem 3.2.7.]. A direct application of the framework of synthethic topology to represented spaces was performed by COLLINS [19, 20] in 2010, several of the result presented here are already present in [20]. Effective compactness has been studied in [16] by BRATTKA and WEIHRAUCH, and in [15] by BRATTKA and PRESSER. [71] by GRUBBA and WEIHRAUCH considers representations of closed, open and compact sets characterized by extremal properties for the restricted case of countably based spaces. Effective topological separation was considered in [69, 70] by WEIHRAUCH. A warning is due regarding a discrepancy in nomenclature: The computable elements of the space of closed subsets introduced here (i.e. the computable closed sets) are called co-c.e. closed sets by WEIHRAUCH and others. Their c.e. closed sets are referred to as overt here (a term coined by TAYLOR, see Section 7), as they as a space lack the structure expected from closed sets. Subsequently, the sets called computably closed in WEIHRAUCH’s terminology are called computable closed and overt here. The observation that the c.e. closed sets lack the closure properties of closed sets spawned various investigations into more restricted setting remedying the issue. An example of this is ZIEGLER’s work on representations for regular closed subsets of a Euclidean space [73]. A second example, somewhat 3Hence the chance of the term of choice from KREITZ’ and WEIHRAUCH’s Theory of Representations [37] to the present paper’s Theory of Represented Spaces. 4The form of admissibility presented in this paper corresponds to admissibility w.r.t. some topology. Schroder¨ also studied admissibility w.r.t some variants of limit spaces. A. Pauly 3 further removed from representations, is the observation that computable closed sets (i.e. co-c.e. closed) satisfying some topological criterion have to be computably overt (i.e. c.e. closed) as made by MILLER and ILJAZOVIC´ [42, 41, 32, 33]. As the attribution of the definitions and results is non-trivial due to parallel independent developments using slightly different formal frameworks, the discussion of the intellectual pedigree follows separately each section. In addition to work cited there, several observations presumably are folklore. This paper is originally based on [46, Chapter 3], the PhD thesis of the author. 3 The foundations The central notion is the represented space, which is a pair X = (X;dX ) of a set X and a partial surjection 5 dX :⊆ f0;1gN ! X ( ). A multi-valued function between represented spaces is a multi-valued function between the underlying sets. For f :⊆ X ⇒ Y and F :⊆ f0;1gN ! f0;1gN, we call F a realizer of f (notation F ` f ), iff dY (F(p)) 2 f (dX (p)) for all p 2 dom( f dX ). A map between represented spaces is called computable (continuous), iff it has a computable (continuous) realizer. Similarly, we call a point ∼ x 2 X computable, iff there is some computable p 2 NN with dX(p) = x. We write X = Y if the two spaces are computably isomorphic, i.e. if there is a bijection f : X ! Y that is computable and has a computable inverse. For our purposes, there is no need to distinguish computably isomorphic spaces. Warning: A priori, the notion of a continuous map between represented spaces and a continuous map between topological spaces are distinct and should not be confused! We consider two categories of represented spaces, one equipped with the computable maps, and one equipped with the continuous maps. We call the resulting structure a category extension (cf. [45, 47]), as the former is a subcategory of the latter, and shares its structure (products, coproducts, exponentials) as we shall see next. In general, all our results have two instances, one for the computable and one for the continuous maps, with the proofs being identical.
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